Verify:   `csc(theta) - cot (theta) = 1/csc(theta)+cot(theta) `

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The identity `csc(theta) - cot (theta) = 1/csc(theta)+cot(theta)` has to be verified.

The left hand side

`csc(theta) - cot (theta)`

=> `1/sin theta - cos theta/sin theta`

=> `(1 - cos theta)/sin theta` ...(1)

The right hand side

`1/csc(theta) + cot(theta)`

=> `sin theta + cos theta/sin theta`

=>` (sin^2 theta + cos theta)/sin theta` ...(2)

(1) and (2) are equal if `1 - cos theta = sin^2 theta + cos theta`

=>` 2*cos theta = 1 - sin ^2 theta`

=> `2*cos theta = cos^2 theta`

=> `cos theta = 0 or cos theta = 2`

This show that` csc(theta) - cot (theta) = 1/csc(theta)+cot(theta)` is true only if `cos theta = 0` . It is not an identity.

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